Conservation of Energy

Karen Tennennhouse

Definitions

1. A system means any group of objects we choose to examine. For a given system, we say that a force is internal iff it is exerted BY an object IN the system (on another object in the system). We say a force is external iff it is exerted BY an object which is NOT IN the system (on an object in the system).
2. The definition of Work done by a constant force is:
${\displaystyle Work=(F_{parallel})(\Delta s)=F(\Delta s)Cos\theta }$
which is also written as
${\displaystyle {\vec {F}}.{\vec {\Delta s}}}$

The work measures the energy transfer caused by this force....... transfer from one object to another or from one energy form to another.
[The more general definition of work is:
${\displaystyle Work=\int {\vec {F}}.{\vec {ds}}}$.
For a one dimensional case, with ${\displaystyle {\vec {F}}}$ parallel to the path, this integral can be found by taking the area of a graph of F versus ‘s’.
In this course, we will use this fact only once, to get our formula for elastic potential energy.]
3. Recall, the units of energy and work are ${\displaystyle kg.m^{2}/s^{2}}$, called Joules.
Equivalently, one Joule is the work done by one Newton of force acting parallel to one meter of displacement,i.e.
1 Joule = 1 N-m
(Caution: although Torques are also measured in N-m, they are not the same thing.)
Another unit of energy is the calorie (c).
1 cal = 4.18 Joules.
The food calorie (C) is 1 Cal = 1000 cals.

Some forms of Energy

• Kinetic Energy (KE): A moving object has more energy than the same object had when at rest; the difference is called the kinetic energy. For speeds much less than the speed of light (including all problems in this course)

${\displaystyle KE={\frac {1}{2}}mv^{2}}$

• Heat(Thermal Energy): In any object or substance, the microscopic particles, (atoms, electrons, etc.) are always in random motion or vibration. For the substance to be warmer (higher temperature) actually means that the average KE of these random microscopic motions is larger than when the substance is cooler.

• Light carries energy.

• Sound carries energy.

• Mass is energy! The famous equation ${\displaystyle E=mc^{2}}$ tells us how much energy is equivalent to an amount ‘m’ of mass. c is the speed of light, ${\displaystyle 3x10^{8}}$ m/s , so one kg of matter contains an enormous ${\displaystyle 9x10^{16}}$ J of energy.

• Chemical Energy. This is a somewhat sloppy but still useful term. When we encounter chemical energy in a mechanics problem, most often it is either the chemical energy of fuels (wood, gasoline, etc.) or chemical energy of a person’s body being used up (to enable the person to move, keep warm, etc. or to do work on other objects.)

• Gravitational Potential Energy (GPE) is energy which has been “borrowed” when we did work against gravity, and which gravity will “pay back” if the object returns to its earlier position. Near the surface of the Earth, whenever an object moves upwards by an amount ${\displaystyle \Delta h}$, its GPE increases by an amount ${\displaystyle \Delta GPE=mg\Delta h}$. Similarly, if the object moves down, it loses GPE.

• Elastic Potential Energy is energy which has been “borrowed” when we did work against a spring or other elastic force. We will see that Elastic ${\displaystyle PE={\frac {1}{2}}k(\Delta l)^{2}}$ where k is the constant of the spring, and ${\displaystyle \Delta l}$ is the amount of stretch or compression, measured from the relaxed position of the spring.

• Electrostatic Potential Energy is energy which was borrowed when we did work against electrical forces (for example, by pushing two same-sign charges closer together.) You’ll learn more about electrostatic PE next year, in Physics NYB.

We define the Total Mechanical Energy (written TME or just ME) as the sum of ( KE + all forms of potential energy.)

But what is “potential energy”?
Why do we talk about “gravitational PE”, but NOT “frictional PE” or “muscle PE”?

• Some forces, such as gravity, are “honest borrowers.” That is, whenever we seem to lose energy by doing work against gravity during a certain trip, we know that gravity will pay back exactly the same amount of energy if the object does the reverse trip back to its previous position. In fact, if no other forces are acting on the return trip, gravity will pay back this loan in the form of kinetic energy.
• A force with this property is called a conservative force. (Notice, not the same thing as conserved or conservation.)
• Each kind of conservative force has a corresponding form of potential energy

Which forces are conservative?

Examples of Conservative Forces Examples of NON-Conservative Forces
• Gravity, i.e. Weight
• Elastic tensions and compressions, i.e. those which obey Hooke’s law.
• The electrostatic (coulomb) force between electric charges.
• A very few others, that you’ll meet in future courses.
Almost everything that’s not in the left column. In particular:
• Non-elastic tensions and compressions (e.g. ropes, normal force by most solids, etc.)
• Forces by muscles
• All forms of friction

Roughly speaking, the mechanical forms of energy (KE and various potential energies) are related to “organized” motion of biggish objects. Non-mechanical forms, such as heat and chemical energy, are related to disorganized, random, microscopic events. The branch of Physics called Thermodynamics studies the interesting laws related to these ideas.

Law of Conservation of Energy

The most general form of the energy law is:

IF the net work done on a system by external forces is zero,
THEN the total energy of the system (sum of all forms) remains constant.


In our present course, we mostly use this law to solve simple problems, and for qualitative questions.

There are two more specialized laws, which deal with the sub-group of mechanical energies.

In this course, we’ll solve most large, complex, energy problems using these two laws:

IF the net work done by non-conservative forces is zero, THEN total mechanical energy stays constant.

You can abbreviate this law as:

If ${\displaystyle Work_{NON-CONS}=0}$, then

${\displaystyle ME_{i}=ME_{f}=ME}$ at all times.

• In general,the change in the mechanical energy equals the net work done by non-conservative forces,i.e.

${\displaystyle ME_{f}-ME_{i}=Work_{NON-CONS}}$

• You can probably see that the second law includes the first one as a special case.

Method for solving Large Energy Problems, using the mechanical energy laws

1. Choose the SYSTEM.
2. CHECK: Does the net work done by non-conservative forces add up to zero?
At this step we need to think carefully about each force which is acting on the system (both external and internal.)
• Is this force conservative or not?
• If not, does it do non-zero work?
You don’t have to write down all these thinking steps, but you must think them through carefully.
3. Write the LAW you plan to use:
If yes to B, plan to use ${\displaystyle ME_{i}=ME_{f}=ME}$ at all times
If no to B, plan to use ${\displaystyle ME_{f}-ME_{i}=Work_{NON-CONS}}$
4. Large, clear must include:
• Show and clearly label the different positions of the object.
Label what you are calling the initial and final (i and f), or label positions A, B, C, etc. if there are several important ones.
• If the problem will involve GPE, choose and label the “zero-point”, where you will consider h = 0.
If you plan to use the formula ${\displaystyle \Delta {GPE}=mg\Delta h}$ you must choose +ve axis upwards for h.
• Label with symbols the relevant distances and angles that you will use in your equations.
(for example, ${\displaystyle h_{f}}$ or ${\displaystyle \Delta l}$ or ${\displaystyle \Delta s}$ etc.)
• NB: This is NOT an isolation diagram.

5. Write out the EQUATION applying your chosen law to this problem.
• This is where you put in the specific energy forms or etc. which are present in this problem.
• Use careful symbols and subscripts.
• If you’re using ${\displaystyle ME_{f}-ME_{i}=Work_{NON-CONS}}$, your right-hand side may have one or several terms. The thinking you did for the “check” step will help you now.
• Make sure that any symbols for unknowns in your equation have been labelled in your main diagram.
6. Substitute formulas and values. Solve. Express final answers.

• When sliding (kinetic) friction does work, it usually does negative work, making mechanical energy decrease. Some form of energy must increase, and usually it is heat (sometimes sound etc.) So for example, in a certain situation, if work done by friction is – 20 J, then heat or etc produced is + 20 J.

• We define power as the rate of energy transfer.
• Units of power are Joules/sec, usually called Watts.
• Average power by a certain force is ${\displaystyle P_{av}={\frac {\Delta {Energy}}{\Delta {time}}}={\frac {Workdonebythisforce}{\Delta {time}}}}$
• Instantaneous power ${\displaystyle P={\frac {dE}{dt}}}$ (derivative), but in our course, we are only using average or constant power.
• In almost all power problems in this course, just consider the energy changes in ${\displaystyle \Delta t=1second}$.